EXCHANGE RINGS SATISFYING STABLE RANGE CONDITIONS

In this paper, we establish necessary and sufficient conditions for an exchange ring R to satisfy the n-stable range condition. It is shown that an exchange ring R satisfies the n-stable range condition if and only if for any regular a $\in$ R$^n$, there exists a unimodular u $\in$$^n$ R such that au $\in$ R is a group member, and if and only if whenever a$\simeq$$_n$b with a $\in$ R, b $\in$ M$_n$(R), there exist u $\in$ R$^n$, v $\in$$^n$ R such that a = ubv with uv = 1. As an application, we observe that exchange rings satisfying the n-stable range condition can be characterized by Drazin inverses. These also give nontrivial generalizations of [7, Theorem 10], [13, Theorem 10], [15, Theorem] and [16, Theorem. 2A].